Optimal. Leaf size=30 \[ \frac{x^3}{9}-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-\frac{x}{3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0313971, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {4678} \[ \frac{x^3}{9}-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-\frac{x}{3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4678
Rubi steps
\begin{align*} \int x \sqrt{1-x^2} \cos ^{-1}(x) \, dx &=-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-\frac{1}{3} \int \left (1-x^2\right ) \, dx\\ &=-\frac{x}{3}+\frac{x^3}{9}-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0276791, size = 26, normalized size = 0.87 \[ \frac{1}{9} \left (x^3-3 \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-3 x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.15, size = 134, normalized size = 4.5 \begin{align*} -{\frac{i+3\,\arccos \left ( x \right ) }{72} \left ( 4\,i{x}^{3}-4\,\sqrt{-{x}^{2}+1}{x}^{2}-3\,ix+\sqrt{-{x}^{2}+1} \right ) }+{\frac{\arccos \left ( x \right ) +i}{8} \left ( ix-\sqrt{-{x}^{2}+1} \right ) }-{\frac{\arccos \left ( x \right ) -i}{8} \left ( ix+\sqrt{-{x}^{2}+1} \right ) }+{\frac{-i+3\,\arccos \left ( x \right ) }{72} \left ( 4\,i{x}^{3}+4\,\sqrt{-{x}^{2}+1}{x}^{2}-3\,ix-\sqrt{-{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.41756, size = 30, normalized size = 1. \begin{align*} \frac{1}{9} \, x^{3} - \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \arccos \left (x\right ) - \frac{1}{3} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.68538, size = 78, normalized size = 2.6 \begin{align*} \frac{1}{9} \, x^{3} + \frac{1}{3} \,{\left (x^{2} - 1\right )} \sqrt{-x^{2} + 1} \arccos \left (x\right ) - \frac{1}{3} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.00505, size = 37, normalized size = 1.23 \begin{align*} \frac{x^{3}}{9} + \frac{x^{2} \sqrt{1 - x^{2}} \operatorname{acos}{\left (x \right )}}{3} - \frac{x}{3} - \frac{\sqrt{1 - x^{2}} \operatorname{acos}{\left (x \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.08687, size = 30, normalized size = 1. \begin{align*} \frac{1}{9} \, x^{3} - \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \arccos \left (x\right ) - \frac{1}{3} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]